Cuando se tema que se oculten o dilapiden los bienes en que debe ejercitarse una acción real; I Cuando la acción sea personal, siempre que el deudor no tuviere otros bienes que aquellos en que

The long term goal of the Er:Si research program is to realise a telecom wavelength bus for future silicon based quantum computers. With this goal in mind, several of the device requirements details in Section 1.2.1 are currently being addressed. These include the development of high Q optical resonators that are compatible with FinFET technology, and investigations of Stark tunability for the optical transitions [164,165]. In line with this goal, this thesis has focused on the issues of Er spin-readout and coherence, both of which require further investigation.

6.4.1 Improving transition lifetimes and coherence times of single Er ions

Materials such P:Si and Bi:Si have shown great promise for quantum information pro- cessing, with the observation of exceedingly long coherence times for these dopants [16,166]. The same should hold true for the Er:Si devices studied here, if several key improvements are made.

Firstly, the large homogeneous and inhomogeneous optical linewidths measured in Sec- tions 5.4 - 5.5 suggest poor Si crystal quality. This issue could be mitigated with high temperature annealing, which was not performed for reasons detailed in Section 5.5. An- nealing could be used, however, if dopants are implanted into the Si wafer prior to FinFET fabrication. If applied, this change in the manufacturing process should greatly reduce both the homogeneous and inhomogeneous linewidths. With regards to the second point, the number of unique spectroscopic sites should be reduces.

Homogeneous broadening could further be reduced by utilising isotopically enriched I = 0 materials (in particular 28Si) to eliminate nuclear magnetic noise. To put this in

context, the isotopic enrichment of Si yielded a 5000-fold improvement in spin coherence times of individual P:Si dopants [17].

Finally, long coherence times and lifetimes in Er:Si will require the Er electron spins to be polarised, for much the same reasons as Er:YSO. Spin-lattice coupling is the major concern here, as the ultra low Er density should help to suppress both electronic and hyperfine cross-relaxation. If we consider the trigonal Er site for which the 4f-Hamiltonian was determined in Section 5.6, a maximum electronicg-factor of 11.4 is achieved along the axis of the site. This would suggest a field of 4T will be required to reduce the spin-lattice relaxation rate below O(1 min), based on the results achieved with Er:YSO in Chapter

4. If no other dynamics are present, then hyperfine state readout should be achieved in this regime, which would represent an important milestone on the path to an Er based optical-spin bus.

Python Crystal Field

PYCF is a software suite developed by S. Horvath at the University of Canterbury. A detailed description of the software and print-outs of the code can be found in reference [82]. Here it is used primarily for fitting the experimental data from Chapter 5 to the 4f-shell Hamiltonian detailed in Chapter 2. PYCF builds upon the 4f-shell Hamiltonian solver developed by M. Reid in the 1980’s. In particular, the two scripts SLJCALC and JMCALC

from the 1980’s solver form the basis of program, and their python equivalents are critical sub-routines. These two scripts calculate the matrix elements of the following Hamiltonian in theτ, L, S, J and τ, L, S, J, mJ basis, respectively:

H =HF I+HCF +HZ

The components of the first two terms are described in Sections 2.2.1 and 2.2.2. The Zeeman interaction is required because the spectroscopic data is presented as a function of magnetic field. It should also be noted that JMCALCcomes in two variants, selected by

user input. The variant used most often provides a faster ‘truncated’ method of calculat- ing matrix elements. This truncated approach follows the method of Carnall et al. [83], and diagonalises the free-ion and crystal-field components in separate (de-coupled) bases. This approximation leads to order-of-magnitude improvements in fitting speed, while only introducing deviations of a few percent compared with the coupled-basis variant ofJMCALC.

The crystal field library

Once the relevant matrix elements have been determined using SLJCALC and JMCALC, the

next step is to fit the Hamiltonian parameters to the recorded spectra. This fitting is performed by the crystal field library (cfl) subroutine. Based on the CFIT algorithm

developed by M. Reid, cfl takes advantage of modern high-performance computing al-

gorithms [167–170]. As suggested by the title, PYCF is optimised for fitting the crystal 145

field parametersB_qk, as opposed to free-ion parameters. In-fact, the only free-ion paramet- ers which are varied during the fitting process are the Slater terms Fk and the spin-orbit term ζ.

Starting with an initial set of parameters, cfl performs an iterative optimisation by

minimising the least-squares difference between the calculated eigenvalues and experi- mental transition energies. As mentioned previously, the spectroscopic data are taken at multiple magnetic field values. Thus, the cfl routine simultaneously fits the data for

each magnetic field value to a separate Hamiltonian, and fitting-weights can be assigned to the optimisation for these individual Hamiltonians.

As with many optimisation programs, PYCF solutions can converge to different (and non-degenerate) local minima. In order to identify the best global solution, cfl employs

the basinhopping algorithm [171]. Basinhopping employs random steps, whose size and frequency and defined by the user, followed by local minimisation. Then, the metropolis criterion is applied to decide whether the algorithm should move to the newly found local minimum [172]. The bounds on parameter space are defined by user input: if a sufficiently converged solution is not found,cflwill continue to execute until the union of these bounds

Modulator equations

Presented here is the derivation of the modulator power response P. This derivation is based on the physical modulator description presented in Section 3.2. Firstly, the output electric fieldE is described as follows:

E=hα−e−i(ωt−φ−)+βeiϕ−α+ei(ωt+φ+) i eif0t E∗E=hα−ei(ωt−φ−)+βe−iϕ−α+e−i(ωt+φ+) i e−if0t_× h α−e−i(ωt−φ−)+βeiϕ−α+ei(ωt+φ+) i eif0t

If we only consider components at the modulation frequencyω:

E(ω)∗E(ω) =α−β h ei(ωt−φ−+ϕ)_+e−i(ωt−φ−+ϕ)i_−α +β h e−i(ωt+φ+−ϕ)_+ei(ωt+φ+−ϕ)i = 2α−βcos (ωt−φ−+ϕ)−2α+βcos (ωt+φ+−ϕ)

= 2α−β[cos (ωt) cos (−φ−+ϕ) + sin (ωt) sin (−φ−+ϕ)] −2α+β[cos (ωt) cos (φ+−ϕ) + sin (ωt) sin (φ+−ϕ)]

= 2βcos (ωt) [α−cos (−φ−+ϕ)−α+cos (φ+−ϕ)]

+ 2βsin (ωt) [α−sin (−φ−+ϕ)−α+sin (φ+−ϕ)]

The expressions in the square-brackets are now substituted for the termsuandv. This simplifies the form of the equation to:

I = 2βucos (ωt) + 2βvsin(ωt)

The two termsuandvcan then be substituted for trigonometric identities, by assuming 147

the following relations:

sinγ = √ u

u2_+v2

cosγ = √ v

u2_+v2

The intensity can then be re-expressed using the effective ‘angle’γ:

I = 2pu2_+v2_{(sinγcos (ωt) + cosγsin (ωt))}

= 2pu2_+v2_{sin (γ+ωt)}

Expressed in this manner, the RF powerP can be described by the RMS amplitude of the optical intensity, squared:

P ∝u2+v2

Where:

u2+v2= [α−cos (−φ−+ϕ)−α+cos (φ+−ϕ)]2+ [α−sin (−φ−+ϕ)−α+sin (φ+−ϕ)]2

=α2−cos2(−φ−+ϕ) +α2+cos2(φ+−ϕ) +α−α+cos (−φ−+ϕ) cos (φ+−ϕ)

+α2−sin2(−φ−+ϕ) +α2+sin2(φ+−ϕ)−α−α+sin (−φ−+ϕ) sin (φ+−ϕ)

=α2−+α2++α−α+[cos (−φ−+ϕ) cos (φ+−ϕ)−sin (−φ−+ϕ) sin (φ+−ϕ)]

Population lifetime data

Presented in the following two figures are AM spectra recorded in a field of 7T along the D1 optical extinction axis of 167Er:YSO. Each Subfigure was recorded at the temperature indicated, following the method described at the beginning of Section 4.8. The temperature for each measurement was inferred from a low pressure vacuum gauge, in equilibrium with the sample space of the cryostat. Temperature regulation of the sample was achieved with a series of valves between the sample space and vacuum pump.

For temperature range of 1.4 K - 1.8 K, a population model was fit assuming the initial population distribution in Table 4.4. For the measurements at 2.0 K and 2.16 K, the level of initial spin polarisation was described by the following distribution:

Hyperfine state |−7/₂i |−5/₂i |−3/₂i |−1/₂i |+1/₂i |+3/₂i |+5/₂i |+7/₂i

N(t= 0)(%) 0 2 1 3 8 18 27 41

For the measurements at 2.77 K and 3.22 K, the level of initial spin polarisation was described by the following distribution:

Hyperfine state |−7/₂i |−5/₂i |−3/₂i |−1/₂i |+1/₂i |+3/₂i |+5/₂i |+7/₂i

N(t= 0)(%) 6 6 6 6 12 18 18 29

The decrease in initial spin-polarisation with increasing temperature occurs due to the increase in phonons. In particular, the optical spin-pumping achieves equilibrium with the spin depolarising effect of the phonons. Hence the equilibrium level of spin-polarisation reduces with increasing phonon density and temperature.

Frequency (GHz) 0 0.5 1 1.5 Modul ation Respo nse (dB ) -7 -6 -5 -4 -3 -2 -1 0 Frequency (GHz) 0 0.5 1 1.5 Modul ation Respo nse (dB ) -7 -6 -5 -4 -3 -2 -1 0 1.4 K 10 seconds 20 minutes 50 minutes model (50 mins) Frequency (GHz) 0 0.5 1 1.5 Modul ation Respo nse (dB ) -7 -6 -5 -4 -3 -2 -1 0 2.0 K 10 seconds 5 minutes 10 minutes 20 minutes model (20 mins)

a)

b)

c)

1.8 K 10 seconds 10 minutes 20 minutes 80 minutes model (80 mins)

Figure C.1: Amplitude modulation spectra of the ∆mI = 0 and +1 optical absorption bands

in a field of 7 T along the D1 axis in167Er:Y2SiO5. Sub-figures a), b) and c) were recorded at helium temperatures of 1.4 K, 1.8 K and 2.0 K respectively. The nuclear spin population is initially pumped into the |+7/2ihyperfine ground state, and the legend entries indicate the time delay of the recorded spectra after spin-pumping. The black dashed traces indicate the fit to spectrum for the longest time delay trace, based on the decay rates plotted in Figure 4.15.

Frequency (GHz) 0 0.5 1 1.5 Modula tion R esponse (dB) -7 -6 -5 -4 -3 -2 -1 0 0 0.5 1 1.5 -7 -6 -5 -4 -3 -2 -1 0 Modula tion R esponse (dB) Frequency (GHz) 0 0.5 1 1.5 -7 -6 -5 -4 -3 -2 -1 0 Modula tion R esponse (dB) Frequency (GHz)

a)

b)

c)

10 seconds 2 minutes 8 minutes model (8 mins) 2.16 K 3 seconds 5 seconds 10 seconds model (10 secs) 2.77 K 5 seconds 10 seconds 20 seconds model (20 secs) 3.22 K

Figure C.2: Amplitude modulation spectra of the∆mI = 0and+1optical absorption bands in a

field of 7 T along theD1 axis in167Er:Y2SiO5. Sub-figuresa),b)andc)were recorded at helium temperatures of 2.16 K, 2.77 K and 3.22 K respectively. The nuclear spin population is initially pumped into the |+7/2i hyperfine ground state, and the legend entries indicate the time delay of the recorded spectra after spin-pumping. The black dashed traces indicate the fit to spectrum for the longest time delay trace, based on the decay rates plotted in Figure 4.15.

Holeburning of the hyper-polarised

ensemble

With the nuclear spin ensemble hyper-polarised, a holeburning spectrum was subsequently recorded. This measurement aimed to create a narrow anti-hole with a low absorbing background, as this type of spectral feature is useful for quantum memory applications. This measurement also confirmed the hyperfine energy structure that was investigated in Section 4.5.

For this experiment the EOM sideband was used to burn a single hole in the centre of the|+7_{/2i ↔ |}+7_/2itransition (the downwards pointing arrow). The hole was burnt with a weak 100 µW pulse for 100 ms, to mitigate hole broadening.

Figure D.1 shows the hole-burning spectrum acquired by AM spectroscopy. The image of the spectral hole in centre of the |+7/₂i ↔ |+5/₂i transition indicates that about one third of the resonant ions had been removed from the |+7/2i ground state1. Three narrow anti-holes were also visible at the frequencies corresponding to the|+5/₂i ↔ |+3/₂i,|+3/₂i ↔ |+1_/2iand_|+1_{/2i ↔ |}−1/₂ioptical transitions. These anti-holes correspond to subsets of the population pumped into the |+5_/2i,|+3_/2iand _|+1_/2iground states, via the_{∆mI =}-1, -2 & -3 optical decay paths. The relative height of the anti-holes presented a means to estimate the oscillators strengths for these ∆mI =-2 & -3 transitions, using Table 4.3. This gave

estimates of 0.7% and 0.3% for the |+7_{/2i ↔ |}+3_/2i and _|+7_{/2i ↔ |}+1_/2i optical transitions, respectively.

1

The hole in the|+7/2i ↔ |+7/2itransition (the downwards pointing arrow) could not be used to infer this value. The large OD caused small modulation-depth for the hole at this wavelength.

                                         ’

Figure D.1: AM spectrum with 95% of the hyperfine ensemble pumped into the |+7/2iground state. Arrow: the frequency where a spectral hole was subsequently burnt, in the centre of the

|+7/2i ↔ |+7/2ioptical transition. Vertical black dashes: The centres of the inhomogeneously broadened 167_{Er optical transitions. Inset: Exploded view of the ∆m}

I = −1 absorption band,

showing side-holes and anti-holes formed by the holeburning. The transitions are labelled according to their corresponding hyperfine ground states.

An additional spectrum of the axial

site

Fitting a crystal-field Hamiltonian generally requires spectra of multiple crystal field levels (Kramers doublets) for a unique solution to be obtained. The spectra developed in Section 5.6 included only optical transitions from the lowest energy Kramers doublet in the I_15/2 state to the lowest doublet in the I13/2 state.

As expected, attempts to determine the Hamiltonian with this limited data gave several non-degenerate solutions. While this meant that more data (spectra) would be required to obtain a unique fit, all the solutions determined at that stage indicated that the next crystal field level in the I13/2 state would be approximately 30 cm−1 (1 THz) higher in energy. This presented a narrow bandwidth (several hundred GHz) over which to develop further optical spectra, in order to identify the secondI13/2 level.

Figure E.1 shows the spectrum developed for this purpose, where four sets of absorption lines were identified. However, only the transitions marked in blue gave a good fit when added to the previous data. This suggested that the secondI13/2 crystal field level for the axial site had been identified.

Figure E.1: A wide-scan absorption spectrum developed over a 0-9T. Absorption lines are ob- served over a range of 800 GHz, each with a unique magnetic field dependence. Only the transitions with blue triangles demonstrated good convergence when added to the previous data for fitting. White lines are a guide to the eye. Data collected by Dr C. Yin.

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